$E_1$ and $E_2$ are events in a probability space satisfying the following constraints:
- $\operatorname{Pr}(E_1)=\operatorname{Pr}(E_2)$
- $\operatorname{Pr}(E_1\cup E_2)=1$
- $E_1$ and $E_2$ are independent.
The probability of $E_1$ is ...(a) $0$, (b) $1/4$, (c) $1/2$, (d) $1$
I think the answer should be (d) 1
Reasoning.
- E1 and E2 are equally likely.
- Sum of their probability is 1. This is possible if and only if both of their probabilities are either 1(edit: if the events are independent) or 0.5( edit: if these are dependent and exhaustive events)
- E1, E2 are independent events. This implies that they both doesn't belong to same same space.
Hence E1 and E2 are certain events with probability 1.
Please correct me if my reasoning or answer is wrong.